The Lorenz attractor and the problem of turbulence

1976 ◽  
pp. 146-158 ◽  
Author(s):  
David Ruelle
Keyword(s):  
Lorenz Attractor

scholarly journals On the Multifractal Character of the Lorenz Attractor

1992 ◽  
Vol 87 (5) ◽  
pp. 1107-1118 ◽  
Author(s):  
R. Dominguez-Tenreiro ◽  
L. J. Roy ◽  
V. J. Martinez
Keyword(s):  
Lorenz Attractor

A Study On Methods for Determining Phase Space Reconstruction Parameters

2021 ◽  
Author(s):  
Shihui Lang ◽  
Zhu Hua ◽  
Guodong Sun ◽  
Yu Jiang ◽  
Chunling Wei
Keyword(s):  
Time Delay ◽  
Phase Space ◽  
Correlation Dimension ◽  
Nearest Neighbor ◽  
Correlation Method ◽  
Phase Space Reconstruction ◽  
Lorenz Attractor ◽  
Embedding Dimension ◽  
Phase Trajectories ◽  
Auto Correlation

Abstract Several pairs of algorithms were used to determine the phase space reconstruction parameters to analyze the dynamic characteristics of chaotic time series. The reconstructed phase trajectories were compared with the original phase trajectories of the Lorenz attractor, Rössler attractor, and Chens attractor to obtain the optimum method for determining the phase space reconstruction parameters with high precision and efficiency. The research results show that the false nearest neighbor method and the complex auto-correlation method provided the best results. The saturated embedding dimension method based on the saturated correlation dimension method is proposed to calculate the time delay. Different time delays are obtained by changing the embedding dimension parameters of the complex auto-correlation method. The optimum time delay occurs at the point where the time delay is stable. The validity of the method is verified through combing the application of correlation dimension, showing that the proposed method is suitable for the effective determination of the phase space reconstruction parameters.

The Outstructure of the Lorenz Attractor

2020 ◽  
pp. 23-32
Author(s):  
Ralph H. Abraham ◽  
Christopher C. Shaw
Keyword(s):  
Lorenz Attractor

scholarly journals Chaotic Convection in a Ferrofluid with Internal Heat Generation

CFD letters ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 62-74
Author(s):  
Nor Halawati Senin ◽  
Nor Fadzillah Mohd Mokhtar ◽  
Mohamad Hasan Abdul Sathar
Keyword(s):  
Heat Generation ◽  
Mathematical Formulation ◽  
Internal Heat ◽  
Boussinesq Approximation ◽  
Internal Heat Generation ◽  
Lorenz Attractor ◽  
Nonlinear Stability Analysis ◽  
Layer System ◽  
Chaotic Convection ◽  
The Impact

The nonlinear stability analysis of a ferrofluid layer system is formulated mathematically. This system considered the upper and lower free isothermal boundary with the system heated from below. A mathematical formulation is produced to study the behaviour of the chaotic convection in a ferrofluid layer system using Galerkin truncated expansion. The Boussinesq approximation is opted with the existence of internal heating and the magnetic number. It is found that the transition to chaos in this present study is identical to the Lorenz attractor and thus validate the method and analysis of this study. The impact of elevating the internal heat generation is found to hasten the instability of the system and as for the magnetic number, at M1 = 2.5 the homoclinic bifurcation occurs and thus accelerates the convection process.

Evaluating the Effects of Size and Precision of Training Data on ANN Training Performance for the Prediction of Chaotic Time Series Patterns

2022 ◽  
pp. 266-282
Author(s):  
Lei Zhang
Keyword(s):  
Time Series ◽  
Open Loop ◽  
Chaotic Time Series ◽  
Training Data ◽  
Acquisition Time ◽  
Lorenz Attractor ◽  
Ann Model ◽  
Training Performance ◽  
Hidden Layer ◽  
Input Delays

In this research, artificial neural networks (ANN) with various architectures are trained to generate the chaotic time series patterns of the Lorenz attractor. The ANN training performance is evaluated based on the size and precision of the training data. The nonlinear Auto-Regressive (NAR) model is trained in open loop mode first. The trained model is then used with closed loop feedback to predict the chaotic time series outputs. The research goal is to use the designed NAR ANN model for the simulation and analysis of Electroencephalogram (EEG) signals in order to study brain activities. A simple ANN topology with a single hidden layer of 3 to 16 neurons and 1 to 4 input delays is used. The training performance is measured by averaged mean square error. It is found that the training performance cannot be improved by solely increasing the training data size. However, the training performance can be improved by increasing the precision of the training data. This provides useful knowledge towards reducing the number of EEG data samples and corresponding acquisition time for prediction.

A topological invariant of the Lorenz attractor

1992 ◽  
Vol 47 (2) ◽  
pp. 221-223 ◽  
Author(s):  
N E Klinshpont
Keyword(s):  
Topological Invariant ◽  
Lorenz Attractor

Use of Experimental Data for an Efficient Description of Turbulent Flows

1990 ◽  
Vol 43 (5S) ◽  
pp. S240-S245 ◽  
Author(s):  
N. Aubry
Keyword(s):  
Experimental Data ◽  
Turbulent Flows ◽  
Turbulence Modeling ◽  
Stokes Equations ◽  
Original System ◽  
Navier Stokes ◽  
Lorenz Attractor ◽  
Wall Region ◽  
Lorenz Equations ◽  
Chaotic Solution

The proper orthogonal decomposition (POD), also called Karhunen-Loe`ve expansion, which extracts ‘coherent structures’ from experimental data, is a very efficient tool for analyzing and modeling turbulent flows. It has been shown that it converges faster than any other expansion in terms of kinetic energy (Lumley 1970). First, the POD is applied to the chaotic solution of the Lorenz equations. The dynamics of the Lorenz attractor is reconstructed by only the first three POD modes. In the second part of this paper, we show how the POD can be used in turbulence modeling. The particular case studied is the wall region of a turbulent boundary layer. In this flow, the velocity field is expanded into POD modes in the normal direction and Fourier modes in the streamwise and spanwise directions. Dynamical systems are obtained by Galerkin projections of the Navier Stokes equations onto the different modes. Aubry et al. (1988) applied the technique to derive and study a ten dimensional representation which reproduced qualitatively the bursting event experimentally observed. It is shown that streamwise modes, absent in Aubry et al.’s model, participate to the bursting events. This agrees remarkably well with experimental observations. In both examples, the dynamics of the original system is very well recovered from the contribution of only a few modes.

The Problem of the Attractor

2005 ◽  
Vol 22 (5) ◽  
pp. 45-65 ◽  
Author(s):  
Adrian Mackenzie
Keyword(s):  
World View ◽  
Deterministic Chaos ◽  
Theoretical Work ◽  
Complexity Science ◽  
Theory Building ◽  
Lorenz Attractor ◽  
Complexity Sciences ◽  
Social Sciences And Humanities ◽  
Recent Theoretical Work ◽  
History Of

Contemporary complexity sciences claim a literal, non-metaphorical applicability to physical, economic, social and cultural events. They envision the development of a general social or historical physics. Conversely, in the social sciences and humanities, complexity sciences have been typically treated as a source of new metaphors or tropes to be used in theory-building. Can there be a critical social or historical physics that is not a world-view and that does not treat science as a source of metaphors? The Lorenz attractor figures centrally in the history of complexity science as a popular image of ‘deterministic chaos’ and the ‘butterfly effect’, as an indication of how far complexity science has progressed in the last two decades, and, as this article argues, as an event whose multiplicity of interpretations attests to the problem it raises, the problem of generality associated with complexity. Via the Lorenz attractor, the article examines three attempts to treat complexity non-metaphorically in recent theoretical work (Delanda; Massumi; Stengers). In these accounts, the attractor performs several different functions. It forms part of a re-engineered concept of multiplicity, it helps conceptualize feeling or sensitivity, and it raises the general problem of practice in theory-building.

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